LONDON—Prime Minister Theresa May’s Brexit plan narrowly survived a series of votes in parliament on Tuesday, a struggle that casts doubt on whether she can negotiate an exit deal with the European Union that is acceptable to British lawmakers.

The skirmishes in parliament, led by the pro- and anti-EU wings of Mrs. May’s ruling Conservatives, highlight how Mrs. May’s efforts to bury party divisions over Brexit have instead inflamed them. That undermines the chances for the smooth and orderly withdrawal that businesses say...